To find the area and perimeter of the triangular region defined by the points A(1, 6), B(-3, -3), and C(6, 1), we first need to calculate both the area and perimeter, taking into account the unit scale provided (where each unit on the coordinate plane represents 10 miles).
Step 1: Calculate the Area of Triangle ABC
The formula for the area of a triangle with vertices at \((x_1, y_1)\), \((x_2, y_2)\), and \((x_3, y_3)\) is given by:
\[ \text{Area} = \frac{1}{2} \left| x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) \right| \]
Substituting the coordinates for points A, B, and C into the formula:
- \(A(1, 6)\) -> \( (x_1, y_1) = (1, 6) \)
- \(B(-3, -3)\) -> \( (x_2, y_2) = (-3, -3) \)
- \(C(6, 1)\) -> \( (x_3, y_3) = (6, 1) \)
Calculating the area:
\[ \text{Area} = \frac{1}{2} \left| 1 (-3 - 1) + (-3)(1 - 6) + 6(6 - (-3)) \right| \] \[ = \frac{1}{2} \left| 1 \cdot (-4) + (-3)(-5) + 6 \cdot 9 \right| \] \[ = \frac{1}{2} \left| -4 + 15 + 54 \right| \] \[ = \frac{1}{2} \left| 65 \right| = \frac{65}{2} = 32.5 \]
Since each unit represents 10 miles, the area in square miles is:
\[ \text{Area} = 32.5 \text{ units}^2 \times (10 \text{ miles/unit})^2 = 32.5 \times 100 = 3250 \text{ square miles} \]
Step 2: Calculate the Perimeter of Triangle ABC
The perimeter is found by calculating the lengths of the sides of the triangle using the distance formula:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
Calculating each side length:
- Side AB:
\[ AB = \sqrt{(1 - (-3))^2 + (6 - (-3))^2} = \sqrt{(1 + 3)^2 + (6 + 3)^2} = \sqrt{4^2 + 9^2} = \sqrt{16 + 81} = \sqrt{97} \]
- Side BC:
\[ BC = \sqrt{(-3 - 6)^2 + (-3 - 1)^2} = \sqrt{(-9)^2 + (-4)^2} = \sqrt{81 + 16} = \sqrt{97} \]
- Side AC:
\[ AC = \sqrt{(1 - 6)^2 + (6 - 1)^2} = \sqrt{(-5)^2 + (5)^2} = \sqrt{25 + 25} = \sqrt{50} \]
The total perimeter is:
\[ \text{Perimeter} = AB + BC + AC = \sqrt{97} + \sqrt{97} + \sqrt{50} = 2\sqrt{97} + \sqrt{50} \]
Calculating these values:
\[ 2\sqrt{97} \approx 2 \times 9.85 \approx 19.7 \] \[ \sqrt{50} \approx 7.07 \] Thus, \[ \text{Perimeter} \approx 19.7 + 7.07 \approx 26.77 \text{ (in units)} \]
Converting to miles, since each unit is 10 miles:
\[ \text{Perimeter} \approx 26.77 \times 10 = 267.7 \text{ miles} \]
Conclusion:
The area discovered is \(3,250\) square miles, which aligns with option C.
The perimeter of the triangle is approximately \(268\) miles, which corresponds with option D.
Thus, the correct answers are:
- C. The area is 3,250 square miles.
- D. The perimeter is approximately 268 miles.